System and methods for adaptive equalization for optical modulation formats

ABSTRACT

A method for modifying the performance of an adaptive equalizer in a receiver is provided. A carrier wave comprising a first polarization state and a second polarization state, wherein there is a correlation between the first polarization state and the second polarization state, is received. The first polarization state and the second polarization state are demultiplexed to generate two respective constellations. A first expected value based on the first constellation, and a second expected value based on the second constellation, are calculated. An adaptive equalizer receives a first and second signals associated with the respective polarization states. The adaptive equalizer generates outputs based on the signals. These outputs are used to generate error values. Feedback is input into the adaptive equalizer, wherein the feedback is based on the error values.

This application is a continuation of U.S. patent application Ser. No.13/307,429, filed on Nov. 30, 2011, and issued on Feb. 2, 2016 as U.S.Pat. No. 9,252,988, the disclosure of which is incorporated by referenceherein in its entirety.

TECHNICAL FIELD

The present disclosure relates generally to optical communications, andmore particularly to adaptive equalization for modulated opticalsignals.

BACKGROUND

The popularity of multimedia communications services over packet datanetworks, such as the Internet, continues to grow. Consequently, thedemand for higher capacity in core data transport networks continues togrow. For service providers, core data transport networks are opticalnetworks based on fiber optic technology. To increase the capacity ofoptical networks, advanced signal modulation techniques, such asquadrature amplitude modulation (QAM) and quadrature phase shift keying(QPSK) have been developed. The push for higher spectral efficiencies tolower the cost per transmitted bit and the concern about exhausting thefiber bandwidth has focused much recent research work on multi-level,multi-dimensional modulation formats to achieve the ultimate capacity ina single fiber.

Digital coherent detection has proven to be an effective technique fordetecting and demodulating the received optical signals based onmulti-level, multi-dimensional modulation formats. Although progress hasbeen made, due to the increased optical-signal-to-noise ratio (OSNR)requirements, the reach of these multi-level, multi-dimensionalmodulation formats is clearly a concern. Thus, power efficientmodulation formats, those having a low required signal-to-noise ratioper bit for a given bit-error-ratio, have also received attention, withsignificant focus on four-dimensional optimized formats (i.e. thoseusing both quadratures and polarization components of theelectromagnetic fields). Power-efficient modulation formats are offundamental importance in optical communications because they providethe ultimate sensitivity limit for the optical channel. Such modulationformats also have practical importance because they enable increasednonlinear tolerance, and therefore the potential for ultra long-haultransmission.

SUMMARY

In accordance with an embodiment, a method and system for modifying theperformance of an adaptive equalizer in a receiver is provided. Acarrier wave comprising a first polarization state and a secondpolarization state, wherein there is a correlation between the firstpolarization state and the second polarization state, is received. Afirst expected value based on the first polarization state and a secondexpected value based on the second polarization state are calculated. Afirst output based on a first signal associated with the firstpolarization state and a second output based on a second signalassociated with the second polarization state are equalized. A firsterror based on the first expected value and the first output and asecond error based on the second expected value and the second outputare calculated. A feedback signal is generated based on the first error,the first output and the correlation.

In some embodiments, the first output and the second output areequalized by an adaptive equalizer that comprises a plurality of finiteimpulse response filters. In certain embodiments, a particularfinite-impulse response filter of the plurality of finite-impulseresponse filters is modified based on the feedback based on the firsterror, the first signal and the complex conjugate of the first signal.In certain embodiments, the first output is determined based on thefirst signal, the tap weight, and the second signal.

These and other advantages of the present disclosure will be apparent tothose of ordinary skill in the art by reference to the followingDetailed Description and the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic of a generic optical communications system;

FIG. 2a shows a schematic illustration of a digital coherent receiver inaccordance with an embodiment;

FIG. 2b shows a schematic illustration of a digital coherent receiver inaccordance with an embodiment;

FIG. 3 shows a schematic illustration of a digital coherent receiverthat includes a 2×2 adaptive equalizer in accordance with an embodiment;

FIG. 4a and FIG. 4b show recovered constellation diagrams for signalsprocessed in accordance with an embodiment;

FIG. 5a and FIG. 5b show recovered constellation diagrams for signalsprocessed in accordance with an embodiment;

FIG. 6 shows a flow chart of the processing steps for signals processedin accordance with an embodiment;

FIG. 7 shows a schematic of a computational system in accordance with anembodiment.

DETAILED DESCRIPTION

In accordance with an embodiment, a system that implements a constrainedmulti-modulus algorithm (MMA) may be used as part of a receiver of asignal formatted according to a four-dimensionally-optimized PS-QPSK(polarization switched-quadrature phase shift keying) modulation format.A receiver operating in accordance with an embodiment may exploit thesignal correlation between the two orthogonal polarizations of thefour-dimensionally-optimized PS-QPSK modulation formatted signal. Thereceiver may de-multiplex the PS-QPSK signal in each of the twoorthogonal polarizations as a five-point constellation (1, j, −1, −j, 0)by minimizing the cross correlation between the two orthogonalpolarization states.

In processing PS-QPSK signals, a constellation diagram may be used toidentify characteristics of a signal. The constellation diagram is asimilar to a scatter plot made in a complex plane where each point inthe scatter plot corresponds to a value of a signal received in aspecific sample of the signal. Generally the scatter plot will havecertain properties that will correspond to the properties of the signalbeing sampled. A constellation diagram may be a representation of asignal modulated by a digital modulation scheme such as quadratureamplitude modulation or phase-shift keying. A constellation diagram maydisplay samples of the signal as a two-dimensional scatter diagram inthe complex plane at symbol sampling instants. A constellation diagramrepresents the possible symbols that may be selected by a givenmodulation scheme as points in the complex plane. Constellation diagramsmay be used to recognize the type of interference and distortion in asignal. The interference and distortion in a signal may be characterizedby determining properties of the associated constellation diagram, forexample a modulus or a radius of the distribution of points in theplots. A particular example of a constellation diagram that may begenerated by a PS-QPSK signal may be five-points in the complex plane atpositions (1, j, −1, −j, 0). Examples of constellation diagrams that maybe generated in accordance with an embodiment of the present disclosureare in FIG. 4 and FIG. 5 of the present disclosure. FIG. 4 and FIG. 5are described in more detail in the foregoing detailed description.

An embodiment of the present disclosure resolves a convergencesingularity problem that may occur in processing PS-QPSK formattedsignals. Embodiments of the disclosed system may also be applicable toother signal coding formats, for example: four-dimensional-coded opticalmodulation formats such as polarization-switched M-ary PSK or M-ary QAM.

FIG. 1 shows a schematic of an optical communications system inaccordance with an embodiment. Multiple transceivers (XCVRs) sendsignals and receive signals via optical transport network 102. Shown arefour representative transceivers, referenced as XCVR 1 104, XCVR 2 106,XCVR 3 108, and XCVR 4 110, respectively. In some communicationssystems, transport network 102 may include optical components. In othercommunications systems, transport network 102 may include opticalcomponents, electronic components, or optoelectronic components. Thetransport medium in transport network 102 may be optical fiber, anelectrical conductor, air or another appropriate medium.

FIG. 2a shows a schematic of an example of a digital coherent receiver232 that may be used to receive four dimensionally coded opticalmodulation formats in accordance with an embodiment. For example,digital coherent receiver may correspond to a particular example of areceiver that is part of a transceiver, such as XCVR 1 as shown inFIG. 1. Input beam 202, with wavelength λ, is received from opticaltransport network 102 (see FIG. 1). Input beam 202 may, for example bean optical carrier wave modulated with information symbols on twoorthogonal polarizations, e.g. the X and Y polarization states. Thedigital coherent receiver 232 receives input beam 202 and determines theamplitude, frequency, and phase of the input beam 202. In an embodiment,input beam 202 may be a modulated optical carrier wave that is processedby digital coherent receiver 232 to recover and decode the informationsymbols. Once received at digital coherent receiver 232, input beam 202may be transmitted into a polarization diverse 90 degree hybrid 210which serves as a polarization and phase-diverse coherent mixer. Localoscillator source 262 may generate a reference continuous-wave opticalsignal 264, with wavelength λ (or close to λ). Reference signal 264 maybe transmitted into polarization diverse 90 degree hybrid 210 for use inprocessing and decoding input beam 202.

Polarization diverse 90 degree hybrid 210 mixes input beam 202 withsignal 264 and generates four optical beams 222, 224, 226, and 228,corresponding to the in-phase and quadrature components in the twoorthogonal polarization states (relative to the reference signal 264).The four optical beams 222, 224, 226, and 228 are transmitted intoopto-electrical converters 242, 244, 246, and 248 respectively. Eachopto-electrical converter 242, 244, 246, and 248 receives the respectivecarrier optical beam 222, 224, 226, and 228 and reference optical beam262 and generates corresponding analog electrical signals 252 and 256and quadrature electrical signals 254 and 258. Thus, eachopto-electrical converter (OE) 242, 244, 246, and 248 processes thereceived signal and outputs an analog signal 252, 254, 256, and 258based on its received input optical signal 222, 224, 226, and 228respectively. In an embodiment, OE 242 generates an analog electricalsignal 252 that corresponds to the real-part of the X polarization ofinput beam 202. In an embodiment, OE 244 generates an analog electricalsignal 254 that corresponds to the imaginary-part of the X polarizationof input beam 202. In an embodiment, OE 246 generates an analogelectrical signal 256 that corresponds to the real-part of the Ypolarization of input beam 202. In an embodiment, OE 248 generates ananalog electrical signal 258 that corresponds to the imaginary-part ofthe Y polarization of input beam 202. Each of these signal are digitizedin a respective analog-to-digital converter (ADC) 262, 264, 266, and268, which generate corresponding digitized electrical signals 272, 274,276, and 278. These signals 272, 274, 276, and 278 are transmitted to adigital signal processor (DSP) such as DSP 220 for further processing.In another embodiment shown in FIG. 2b , the polarization-diverse 90degree hybrid may have four pairs of output optical signals 222 and 223,224 and 225, 226 and 227, and 228 and 229: two pairs for the Xpolarization, resulting from mixing the signal's electric field withthat of a local oscillator (LO) with 0 and π, and π/2 and 3π/2 relativephase differences, and two pairs for the Y polarization resulting frommixing the signal's electric field with that of the LO with 0 and π, andπ/2 and 3π/2 relative phase differences. These four pairs of outputoptical signals are coupled to respective pairs of balanced, matchedphoto-detectors 243, 245, 247, and 249 whose electrical outputs aresubtracted to improve the signal-to-noise ratio of the resultingelectrical signals 252, 254, 256, and 258, which are then sent to arespective ADC 262, 264, 266, and 268.

FIG. 3 shows a schematic of an example of DSP 220 that may be used toprocess digitized signals 272, 274, 276, and 278 in accordance with anembodiment. As shown in FIG. 3, digitized signals 272 and 274 thatcorrespond to the X-polarization of input beam 202 are input to a finiteimpulse response filter 322 for static equalization. Digitized signals276 and 278 that correspond to the Y-polarization of input beam 202 areinput to a finite impulse response filter 324 for static equalization.The static equalizer is mainly used to compensate for fiber dispersion.

In an embodiment, each of finite impulse response filters 322 and 324may provide a signal output. Finite impulse response filter 322 providesan output 323 that is input to an adaptive equalizer 330 and finiteimpulse response filter 324 provides an output 325 that is input toadaptive equalizer 330. Adaptive equalizer 330 comprises a finiteimpulse response filter 332, a finite impulse response filter 334, afinite impulse response filter 336, and a finite impulse response filter338. In addition, adaptive equalizer 330 comprises adder 362 and adder364. Each of finite impulse response filters 332, 334, 336, and 338receive feedback and generate outputs. Finite impulse response filters332 and 334 receive signals 323 and 325, respectively, and in response,generate outputs that are input to adder 362, which sums the two signalsand generates output Z_(x), which corresponds to the X-polarization ofinput beam 202. Finite impulse response filters 336 and 338 receivesignals 323 and 325, respectively, and in response, generate outputsthat are input to adder 364, which sums the two signals and generatesoutput Z_(y), which corresponds to the Y-polarization of beam 202.

In an embodiment, output signals Z_(x) and Z_(y) may then be transmittedto carrier recovery and decoding device 380.

Adaptive equalizer 330 receives feedback from an equalization process392 and an equalization process 394. The feedback signals received byadaptive equalizer 330 may be used to vary the respective tap weights offinite impulse response filters 332, 334, 336, and 338 in order toimprove the response of the DSP 220 to signals corresponding to inputbeam 202.

In an illustrative embodiment of the present disclosure, a constrainedMMA may be used to exploit a signal correlation between two orthogonalpolarizations in an incoming beam and de-multiplex a PS-QPSK signal ineach of the two orthogonal polarizations as a five-point constellation(1, j, −1, −j, 0). The demultiplexing may be performed to minimize thecross correlation between the two orthogonal polarization states inincoming beam.

FIG. 6 is a flow chart of a method of providing feedback to an adaptiveequalizer. In accordance with an embodiment, a constrained multi-modulusalgorithm (MMA) may be used to generate the feedback provided toadaptive equalizer 330.

At step 610, a carrier wave comprising a first polarization state and asecond polarization state is received, wherein there is a correlationbetween the first polarization state and the second polarization state.In an illustrative embodiment, input beam 202 comprises the carrier wavethat is received from optical transport network 102 and arrives atdigital coherent receiver 232. Input beam 202 may comprise afour-dimensionally-optimized PS-QPSK modulation format signal. Thesesignals may be processed by elements of a receiver such as thoseillustrated in FIG. 1 and FIG. 2 in order to generate a signal forfurther processing.

At step 620, the first polarization state and the second polarizationstate of input beam 202 are demultiplexed to generate a firstconstellation corresponding to the first polarization state and a secondconstellation corresponding to the second polarization state. In anillustrative embodiment, the two polarization states may bedemultiplexed by a demultiplexing device such as the adaptive equalizer330 in DSP 220 which may record a value of each polarization state foreach phase. Recorded values from a series of input cycles may be used togenerate a constellation diagram corresponding to each polarizationstate.

At step 630, a first expected value is calculated based on the firstconstellation and a second expected value is calculated based on thesecond constellation diagram. Each of the constellation diagrams arepopulated with data collected from each polarization state of theincoming beam, and based upon the data entered into each diagram, aradius or modulus is calculated. The values of the radius or modulus foreach of the X polarization state and the Y polarization state may bestored as the first expected value and the second expected value,respectively. Based on statistics collected from the received signal,data may be collected that corresponds to each of the X-polarization andthe Y-polarization components of input beam 202. These data may be usedto generate a constellation in the complex plane that corresponds to thedegree of distortion and noise in each polarization for incoming beam202. This constellation may, for example, be five points at positions,(1, j, −1, −j, 0) in the complex plane. Based on this data expectedsquare values of the radius/modulus of the constellation for eachpolarization may be calculated. These data may be an indication of thenoisiness of incoming beam 202.

At step 640, a first signal associated with the first polarization stateand a second signal associated with the second polarization state arereceived at an adaptive equalizer such as adaptive equalizer 330. In anembodiment, adaptive equalizer 330 as shown in FIG. 3 comprises fourimpulse response filters and may be of a class of adaptive equalizersthat is called a 2×2 adaptive equalizer.

At step 650, the adaptive equalizer equalizes the received signals togenerate a first output based on the first signal and a second outputbased on the second signal. In an embodiment, the adaptive equalizersuch as adaptive equalizer 330 may be operable to generate a response toan input that varies based on the time varying properties of its inputaccording to feedback it receives. In the foregoing description of theoperation of adaptive equalizer 330 and other components of DSP 220,various mathematical equations are disclosed. One skilled in the artwill understand that these equations may be implemented in softwareexecuted by a processor, hardware or firmware, or in combinationsthereof. Having implemented these equations, a machine may be operatedwhich receives signals as inputs and generates electrical outputs as acomponent of a communication system. In the examples discussed hereinreference is made to specific aspects of adaptive equalizer 330, howeverone skilled in the art will understand how to apply the disclosedtechniques to other systems with comparable features. In the foregoing,the symbols, x(n) and y(n), are used to refer to the inputs to theadaptive equalizer, such as outputs of finite impulse response filters322 and 324 respectively. The outputs of adders 362 and 364, arereferred to with symbols Z_(x)(n) and Z_(y)(n), respectively.

In an embodiment, adaptive equalizer 330 comprises adders 362 and 364that generate outputs, Z_(x)(n) and Z_(y)(n), which are based on x(n)and y(n). The outputs of adders 362 and 364, Z_(x)(n) and Z_(y)(n), maybe calculated mathematically according to equation (1) which specifies amathematical formula for Z_(x)(n) and equation (2) which specifies amathematical formula for Z_(y)(n):

$\begin{matrix}\begin{matrix}{{Z_{x}(n)} = {{\sum\limits_{k = 0}^{K - 1}{{h_{xx}\left( {n,k} \right)}{x\left( {n - k} \right)}}} + {\sum\limits_{k = 0}^{K - 1}{{h_{xy}\left( {n,k} \right)}{y\left( {n - k} \right)}}}}} \\{= {{h_{xx}^{T} \cdot X} + {h_{xy}^{T} \cdot Y}}}\end{matrix} & (1) \\\begin{matrix}{{Z_{y}(n)} = {{\sum\limits_{k = 0}^{K - 1}{{h_{yy}\left( {n,k} \right)}{y\left( {n - k} \right)}}} + {\sum\limits_{k = 0}^{K - 1}{{h_{yx}\left( {n,k} \right)}{x\left( {n - k} \right)}}}}} \\{= {{h_{yy}^{T} \cdot Y} + {h_{yx}^{T} \cdot X}}}\end{matrix} & (2)\end{matrix}$

In equations (1) and (2), h_(xx), h_(xy), h_(yy), h_(yx) refer to tapweights associated with each of finite impulse response filters, such asfinite impulse response filters 332, 334, 336, and 338 respectively.These tap weights may be varied for each finite impulse response filterin order to improve the response of adaptive equalizer 330 to inputsignals. Generally each of the finite response filters associated withh_(xx), h_(xy), h_(yy), h_(yx) respectively may have a frequencydependent response and may be characterized as a complex number. Thus,each of h_(xx), h_(xy), h_(yy), h_(yx) will have an associated Hermiteconjugate, which is denoted: h^(T) _(xx), h^(T) _(xy), h^(T) _(yy),h^(T) _(yx). In accordance with an embodiment, each finite impulseresponse filter will have length K taps.

At step 660, a first error based on the first expected value and thefirst output and a second error based on the second expected value andthe second output are calculated. In an embodiment, the first error andthe second error may be calculated, for example, at equalization process392 and 394. These calculations may be performed based on the expectedsquare values of the radius/modulus of the constellation for eachpolarization calculated in step 630 and the outputs generated by adders362 and 364. In the foregoing equations, R_(x)(n) and R_(y)(n) refer tothe expected square values of the radius/modulus of the constellationassociated with the X- and Y-polarizations, respectively. Using theoutput of equations (1) and (2), and the values of R_(x)(n) andR_(y)(n), equation (3) provides a means for calculating errors cassociated with each of the respective X-polarization and theY-polarization of incoming beam 202.ε_(x)(n)=|Z _(x)(n)|² −R _(x)(n)ε_(y)(n)=|Z _(y)(n)|² −R _(y)(n)  (3)

where equation (3) is constrained by the conditions specified inequation (4).R _(x)(n)=E{|Z _(x)(n)|² +|Z _(y)(n)|²} and R _(y)(n)=0, if |Z_(x)(n)|² >|Z _(y)(n)|²R _(x)(n)=0 and R _(y)(n)=E{|Z _(x)(n)|² +|Z _(y)(n)|²}, if |Z(n)|² <|Z_(y)(n)|²  (4)

-   -   for the case when |Z_(x)(n)|=|Z_(y)(n)|, ε_(x)=ε_(y)=0.

According to an embodiment of the present disclosure, due to thecorrelation of X and Y polarizations, R_(x)(n) and R_(y)(n) arecorrelated and this may be used to generate constraints such as thosespecified in equation (4). In particular, the values for R_(x)(n) andR_(y)(n) are constrained by the instantaneous values of Z_(x)(n) andZ_(y)(n).

At step 670, a feedback signal is input into the adaptive equalizer,wherein the feedback signal is based on the first error and the seconderror and the correlation. In an exemplary embodiment, the feedbacksignal is generated by equalization process 392 and 394. The feedbacksignal provided to adaptive equalizer 330 updates the filtercoefficients by using a steepest gradient algorithm, described byequation (5), where the ε_(x) and Σ_(y) are the feedback errors for thetwo polarizations,

$\begin{matrix}{{\frac{\mathbb{d}\left\langle ɛ_{x}^{2} \right\rangle}{\mathbb{d}h_{xx}} = 0};{\frac{\mathbb{d}\left\langle ɛ_{x}^{2} \right\rangle}{\mathbb{d}h_{xy}} = 0};{\frac{\mathbb{d}{\left\langle ɛ_{y}^{2} \right\rangle}}{\mathbb{d}h_{yy}} = 0};{\frac{\mathbb{d}\left\langle ɛ_{y}^{2} \right\rangle}{\mathbb{d}h_{yx}} = 0};} & (5)\end{matrix}$

According to an embodiment of the present disclosure, the feedbacksignal and hence optimal tap weights, may be calculated based on theconstraints defined in equation (4). Using these constraints, equations(6)-(9) may be used to generate optimal tap weights for finite impulseresponse filters 332, 334, 336, and 338. In particular, in equations(6)-(9), instantaneous values for the tap weights may be used togenerate updated tap weights for each of finite impulse response filters332, 334, 336, and 338 according to the following equations:h _(xx) →h _(xx)+με_(y) Z _(x) ·X   (6)h _(xy) →h _(xy)+με_(x) Z _(x) ·Y   (7)h _(yx) →h _(yx)+με_(y) Z _(y) ·X   (8)h _(yy) →h _(yy)+με_(y) Z _(y) ·Y   (9)

In equations (6)-(9), the symbol μ denotes a convergence parameter thatmay be selected to tune the response speed of the adaptive filter and Xand Y denote the complex conjugates of X and Y, respectively. Thecalculations indicated by equations (6)-(9) may be performed, forexample, as part of equalization process 392 and equalization process394. The calculations associated with equalization processes 392 and 394may be executed on, for example, a computer, an ASIC, a DSP or a similardevice for performing such calculations.

In an embodiment, a means for receiving the input beam 202, such as acarrier wave may be a receiver such as XCVR 1 104, XCVR 2 106, XCVR 3108, and XCVR 4 110. The demultiplexing may be performed by a DSP orother similar device used for demultiplexing a signal. Such a DSP maycomprise a CPU and memory or other components of a computer, whichfunction in order to generate and store data corresponding to aconstellation associated with a particular polarization state of aparticular signal. Receiving a signal and equalizing an output may beperformed by an adaptive equalizer such as a 2×2 adaptive equalizer. Theerror may be calculated by a CPU or other similar device. The feedbacksignal may be input by a signal generator or other similar device.

An embodiment of this disclosure has been used to perform tests todetermine the effectiveness of these techniques. Based on the results ofthese tests, the proposed techniques optimize the equalizer towardminimum cross-correlation for an incoming beam 202 that comprises twopolarizations.

An embodiment of these techniques has been used to experimentally verifyits effectiveness in a recent study involving a 40.5-Gb/s PS-QPSK. FIG.4 and FIG. 5 illustrate recovered constellation diagrams in both X andY-polarization for a back-to-back measurement, and after 10×100 kmstandard single mode fiber (SSMF), wavelength division (WDM)transmission, respectively.

FIGS. 4a and 4b shows a recovered constellation diagrams for an Xpolarization signal (FIG. 4a ) and a Y polarization signal (FIG. 4b ).The distributions shown in these constellations were generated during atest using a 40.5 Gb/s PS-QPSK signal input to an adaptive equalizer inaccordance with an embodiment. In this test, the PS-QPSK signal wasrecovered for back-to-back, with no additional noise loading.

FIGS. 5a and 5b shows a recovered constellation diagrams for an Xpolarization signal (FIG. 5a ) and a Y polarization signal (FIG. 5b ).The distributions shown in these constellations were generated during atest using a 40.5 Gb/s PS-QPSK constellation diagram after 10×100 kmstandard single mode fiber (SSMF), wavelength division (WDM)transmission for 5 dBm/ch launch power and 18.3 dB received OSNR (withloading noise added).

These results have shown that the technique has very robust convergenceperformance and therefore may be used as either an independent blindequalization algorithm for PS-QPSK or as a pre-convergence equalizationalgorithm for PS-QPSK in combination with the use of the well knowndecision-directed least-mean-square algorithm. The techniques disclosedherein may be used to inherently eliminate the convergence singularityproblem and may be useful for the practical implementation of PS-QPSKmodulation format for optical communication systems at 100 Gb/s andhigher rates. The principles applied in the techniques disclosed hereinmay also be applicable to other four-dimensional-coded opticalmodulation formats such as polarization-switched M-ary PSK or M-ary QAM.

FIG. 7 shows an example of a computational system 702 for performing aconstrained multi-modulus algorithm as part of a receiver of a signalformatted according to a four-dimensionally-optimized PS-QPSK modulationformat. One skilled in the art can construct the computational system702 from various combinations of hardware and software (includingfirmware). One skilled in the art can construct the computational system702 from various combinations of electronic components, such as generalpurpose microprocessors, digital signal processors (DSPs),application-specific integrated circuits (ASICs), field-programmablegate arrays (FPGAs), random access memory, and non-volatile read-onlymemory.

Computational system 702 comprises computer 704, which includes adigital signal processor (DSP) 706, memory 708, and data storage device710. Data storage device 710 comprises at least one non-transitory,persistent, tangible computer readable medium, such as non-volatilesemiconductor memory (data storage device 710 can also comprise othernon-transitory, persistent, tangible computer readable medium withsufficiently high data transfer rates).

Computational system 702 further comprises input/output interface 720,which interfaces computer 704 with input/output device 740. Data,including computer executable code can be transferred to and fromcomputer 704 via input/output interface 720. Computational system 702further comprises digital signal interface A 722, which interfacescomputer 704 with digital signal source 742. An example of digitalsignal source 742 is a DSP functional block such as a finite impulseresponse filter 322 that outputs a digital signal x(n). Computationalsystem 702 further comprises digital signal interface A 724, whichinterfaces computer 704 with digital signal receiver 744. An example ofdigital signal receiver 744 is a DSP functional block performing carrierrecovery and decoding 380 that may receive an output digital signal,such as Z_(y)(n).

As is well known, a computer operates in conjunction with computersoftware, which may define the overall operation of the computer and itsassociated applications. DSP 706 may control the overall operation ofthe computer and applications by executing computer program instructionsthat define the overall operation and applications. The computer programinstructions can be stored in data storage device 710 and loaded intomemory 708 when execution of the program instructions is desired. Themethod steps shown in the flowchart in FIG. 6 can be defined by computerprogram instructions stored in memory 708 or in data storage device 710(or in a combination of memory 708 and data storage device 710) andcontrolled by DSP 706 executing the computer program instructions. Forexample, the computer program instructions can be implemented ascomputer executable code programmed by one skilled in the art to performalgorithms implementing the method steps shown in the flowchart in FIG.6. Accordingly, by executing the computer program instructions, the DSP706 executes algorithms implementing the method steps shown in theflowchart in FIG. 6.

The foregoing Detailed Description is to be understood as being in everyrespect illustrative and exemplary, but not restrictive, and the scopeof the inventive concept disclosed herein is not to be determined fromthe Detailed Description, but rather from the claims as interpretedaccording to the full breadth permitted by the patent laws. It is to beunderstood that the embodiments shown and described herein are onlyillustrative of the principles of the present disclosure and thatvarious modifications may be implemented by those skilled in the artwithout departing from the scope and spirit of the disclosure. Thoseskilled in the art could implement various other feature combinationswithout departing from the scope and spirit of the disclosure.

The invention claimed is:
 1. A method, comprising: receiving a carrierwave comprising a first polarization state and a second polarizationstate, wherein a correlation exists between the first polarization stateand the second polarization state; generating a first constellationcorresponding to the first polarization state and a second constellationcorresponding to the second polarization state; calculating a firstexpected value based on the first constellation and a second expectedvalue based on the second constellation; equalizing, by an adaptiveequalizer, a first output based on a first signal associated with thefirst polarization state and a second output based on a second signalassociated with the second polarization state; calculating a first errorbased on a difference between the first expected value and the firstoutput, the first expected value being constrained based on the firstoutput and the second output; calculating a second error based on adifference between the second expected value and the second output, thesecond expected value being constrained based on the first output andthe second output; generating a feedback signal based on the firsterror, the second error and the correlation; and determiningcoefficients for filters of the adaptive equalizer using the feedbacksignal.
 2. The method as recited in claim 1, where the generating afirst constellation corresponding to the first polarization state and asecond constellation corresponding to the second polarization statecomprises: demultiplexing the first polarization state and the secondpolarization state to generate the first constellation and the secondconstellation using a demultiplexer.
 3. The method as recited in claim1, wherein the first constellation comprises a plurality of data pointsassociated with the first polarization state and wherein the data pointsare associated with a diagram in a complex plane, the diagram comprisinga plurality of vertices, each of the respective vertices located at arespective position (1, j, −1, −j, 0) in the complex plane.
 4. Themethod as recited in claim 1, wherein the first expected value iscalculated based on an expected square value of a modulus of the firstconstellation.
 5. The method as recited in claim 1, wherein the firsterror and the second error are calculated using the following equations:ε_(x)(n)=|Z _(x)(n)|² −R _(x)(n),ε_(y)(n)=|Z _(y)(n)|² −R _(y)(n) wherein Z_(x)(n) corresponds to thefirst output, Z_(y)(n) corresponds to the second output, R_(x)(n) is thefirst expected value, and R_(y)(n) is the second expected value.
 6. Themethod as recited in claim 1, wherein the carrier wave is modulatedaccording to at least one of polarization-switched m-ary phase shiftkeying and polarization-switched m-ary quadrature amplitude modulation.7. The method as recited in claim 1, wherein the carrier wave comprisesat least one of an optical carrier wave and a radio-frequency carrierwave.
 8. The method as recited in claim 1, wherein the generating afeedback signal comprises: generating the feedback signal using asteepest gradient algorithm.
 9. The method as recited in claim 1,wherein the adaptive equalizer comprises a plurality of finite impulseresponse filters.
 10. A system, comprising: a receiver for receiving acarrier wave comprising a first polarization state and a secondpolarization state, wherein a correlation exists between the firstpolarization state and the second polarization state; an adaptiveequalizer for: generating a first constellation corresponding to thefirst polarization state and a second constellation corresponding to thesecond polarization state; equalizing a first output based on a firstsignal associated with the first polarization state and a second outputbased on a second signal associated with the second polarization state;a processor; and a memory to store computer program instructions, thecomputer program instructions when executed on the processor cause theprocessor to perform operations comprising: calculating a first expectedvalue based on the first constellation and a second expected value basedon the second constellation; calculating a first error based on adifference between the first expected value and the first output, thefirst expected value being constrained based on the first output and thesecond output; calculating a second error based on a difference betweenthe second expected value and the second output, the second expectedvalue being constrained based on the first output and the second output;and generating a feedback signal based on the first error, the seconderror and the correlation; the adaptive equalizer further fordetermining coefficients for filters of the adaptive equalizer using thefeedback signal.
 11. The system as recited in claim 10, wherein theadaptive equalizer is further for demultiplexing the first polarizationstate and the second polarization state to generate the firstconstellation and the second constellation.
 12. The system as recited inclaim 10, wherein the first constellation comprises a plurality of datapoints associated with the first polarization state and wherein the datapoints are associated with a diagram in a complex plane, the diagramcomprising a plurality of vertices, each of the respective verticeslocated at a respective position (1, j, −1, −j, 0) in the complex plane.13. The system as recited in claim 10, wherein the first expected valueis calculated based on an expected square value of a modulus of thefirst constellation.
 14. The system as recited in claim 10, wherein thefirst error and the second error are calculated using the followingequations:ε_(x)(n)=|Z _(x)(n)|² −R _(x)(n),ε_(y)(n)=|Z _(y)(n)|² −R _(y)(n) wherein Z_(x)(n) corresponds to thefirst output, Z_(y)(n) corresponds to the second output, R_(x)(n) is thefirst expected value, and R_(y)(n) is the second expected value.
 15. Thesystem as recited in claim 10, wherein the carrier wave is modulatedaccording to at least one of polarization-switched m-ary phase shiftkeying and polarization-switched m-ary quadrature amplitude modulation.16. A non-transitory computer readable medium storing computer programinstructions for processing a carrier wave to provide feedback to anadaptive equalizer, the carrier wave comprising a first polarizationstate and a second polarization state, wherein a correlation existsbetween the first polarization state and the second polarization state,the computer program instructions, when executed on a processor, causethe processor to perform operations comprising: calculating a firstexpected value based on a first constellation corresponding to the firstpolarization state and a second expected value based on a secondconstellation corresponding to the second polarization state;calculating a first error based on a difference between the firstexpected value and a first output, the first output being generated byequalizing a first signal associated with the first polarization state,the first expected value being constrained based on the first output andthe second output; calculating a second error based on a differencebetween the second expected value and a second output, the second outputbeing generated by equalizing a second signal associated with the secondpolarization state, the second expected value being constrained based onthe first output and the second output; generating a feedback signalbased on the first error, the second error and the correlation; andtransmitting the feedback signal to the adaptive equalizer fordetermining coefficients for filters of the adaptive equalizer.
 17. Thenon-transitory computer readable medium as recited in claim 16, whereinthe first constellation comprises a plurality of data points associatedwith the first polarization state and wherein the data points areassociated with a diagram in a complex plane, the diagram comprising aplurality of vertices, each of the respective vertices located at arespective position (1, j, −1, −j, 0) in the complex plane.
 18. Thenon-transitory computer readable medium as recited in claim 16, whereinthe first expected value is calculated based on an expected square valueof a modulus of the first constellation.
 19. The non-transitory computerreadable medium as recited in claim 16, wherein the first error and thesecond error are calculated using the following equations:ε_(x)(n)=|Z _(x)(n)|² −R _(x)(n),ε_(y)(n)=|Z _(y)(n)|² −R _(y)(n) wherein Z_(x)(n) corresponds to thefirst output, Z_(y)(n) corresponds to the second output, R_(x)(n) is thefirst expected value, and R_(y)(n) is the second expected value.
 20. Thenon-transitory computer readable medium as recited in claim 16, whereinthe generating a feedback signal comprises: generating the feedbacksignal using a steepest gradient algorithm.